On Ruan's cohomological crepant resolution conjecture for the complexified Bianchi orbifolds

Abstract

We give formulae for the Chen–Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic $3$–space.

The Bianchi groups are the arithmetic groups ${PSL}_2\left(\mathcal{O}\right)$, where $\mathcal{O}$ is the ring of integers in an imaginary quadratic number field. The underlying real orbifolds which help us in our study, given by the action of a Bianchi group on real hyperbolic $3$–space (which is a model for its classifying space for proper actions), have applications in physics.

We then prove that, for any such orbifold, its Chen–Ruan orbifold cohomology ring is isomorphic to the usual cohomology ring of any crepant resolution of its coarse moduli space. By vanishing of the quantum corrections, we show that this result fits in with Ruan’s cohomological crepant resolution conjecture.